This is related to the previous question of how to define a conductor of an elliptic curve or a Galois representation.

What motivated the use of the word "conductor" in the first place?

A friend of mine once pointed out the amusing idea that one can think of the conductor of an elliptic curves as "someone" driving a train which lets you off at the level of the associated modular form.

A similar statement can be made concerning Szpiro's conjecture, which provides asymptotic bounds on several invariants of an elliptic curve in terms of its conductor. Here one might think of the conductor as "someone" who controls this symphony of invariants consisting of the minimal discriminant, the real period, the modular degree, and the order of the Shafarevich-Tate group (assuming BSD).

Was there some statement of this sort which motivated Artin's original definition of the conductor?

Does anyone have a reference for the first appearance of the word conductor in this context?

I believe it is a translation of the German term 'Führer' (which must have led to some awkward conversations between number theorists in the late 1930's and early 1940's). I don't know how and when the German term originated.
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François G. Dorais♦Mar 25 '10 at 3:05

1

"conductor" must be appropriate for $\mho$, if not for MO ;-)
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Noam D. ElkiesJan 29 '13 at 20:47

3 Answers
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It is a translation from the German Führer (which also is the reason that
in older literature, as well as a fair bit of current literature, the conductor
is denoted as f in various fonts). Originally the term conductor appeared in
complex multiplication and class field theory: the conductor of an abelian extension is a certain ideal that controls the situation. Then it drifted off into other
areas of number theory to describe parameters that control other situations.

Of course in English we tend not to think of conductor as a leader
in the strong sense of Führer, but more in a musical sense, so it seems like a weird translation. But back in the
1930s the English translation was leader rather than conductor, at least once: see the review of Fueter’s book on complex multiplication in the 1931 Bulletin of the Amer. Math.
Society, page 655. The reviewer writes in the second paragraph "First there is a careful treatment of those ray class fields whose leaders are multiples of the ideal..."
You can find the review yourself at
http://www.ams.org/bull/1931-37-09/S0002-9904-1931-05214-9/S0002-9904-1931-05214-9.pdf.

I stumbled onto that reference quite by chance (a couple of years ago). If anyone knows other places in older papers in English where conductors were called leaders, please post them as comments below. Thanks!

Concerning Artin's conductor, he was generalizing to non-abelian Galois extensions the parameter already defined for abelian extensions and called the conductor. So it was natural to use the same name for it in the general case.

Edit: I just did a google search on "leader conductor abelian" and the first hit
is this answer. Incredible: it was posted less than 15 minutes ago!

The first time I ever saw a conductor defined is not in the sense mentioned above, but in linear algebra, from Hoffman & Kunze's book. In their chapter on elementary canonical forms they define the conductor of vector $\alpha$ into a subspace $W$ with respect to a linear operator $T$ to be the ideal

$S_T(\alpha;W) = \{ g \in F[x] \mid g(T)\alpha \in W \}$

Where the ambient vector space is over the field $F$. Interestingly, they say that they themselves call this the 'stuffer' ideal (from German, das eistopfende Ideal), but claim that "Conductor" is more commonly used, and add that this term is

"preferred by those who envision a less aggressive operator $g(T)$, gently leading the vector $\alpha$ into $W$."

defined the "Führer" of an order in a number field. [BTW: in German, Führer
does not actually mean a strong leader but rather someone who guides you (as in
tourist guide). But of course . . . ]

Class groups of orders in quadratic number fields are ring class groups, which generalize immediately to ray class groups (Weber); from there the word spread to complex multiplication and class field theory.